In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincaré and $$\beta $$
β
-Logarithmic Sobolev inequalities for measures as a function of this fundamental solution.
In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. That is, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities.
In the setting of Carnot groups, we prove the q−Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carathéodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot groups to prove the p−Talagrand inequality and hypercontractivity. Contents 1. Introduction 1 2. The Carnot Group and the Hamilton-Jacobi Equation 4 3. q−Logarithmic Sobolev Inequality 6 4. p−Talagrand inequality for dµ = e −U (d) Z dλ 10 5. Hypercontractivty of Hamilton-Jacobi solutions for dµ = e −U (d) Z dλ 13 References 14
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